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Properties

Label 18.0.21012633925...1099.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 19^{15}$
Root discriminant $42.56$
Ramified primes $7, 19$
Class number $27$ (GRH)
Class group $[3, 3, 3]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2471, 29932, 167146, 114416, 51290, -46103, -64412, -20046, -231, 8926, 3787, 31, 159, -193, -111, -13, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 7*x^16 - 13*x^15 - 111*x^14 - 193*x^13 + 159*x^12 + 31*x^11 + 3787*x^10 + 8926*x^9 - 231*x^8 - 20046*x^7 - 64412*x^6 - 46103*x^5 + 51290*x^4 + 114416*x^3 + 167146*x^2 + 29932*x + 2471)
 
gp: K = bnfinit(x^18 + 7*x^16 - 13*x^15 - 111*x^14 - 193*x^13 + 159*x^12 + 31*x^11 + 3787*x^10 + 8926*x^9 - 231*x^8 - 20046*x^7 - 64412*x^6 - 46103*x^5 + 51290*x^4 + 114416*x^3 + 167146*x^2 + 29932*x + 2471, 1)
 

Normalized defining polynomial

\( x^{18} + 7 x^{16} - 13 x^{15} - 111 x^{14} - 193 x^{13} + 159 x^{12} + 31 x^{11} + 3787 x^{10} + 8926 x^{9} - 231 x^{8} - 20046 x^{7} - 64412 x^{6} - 46103 x^{5} + 51290 x^{4} + 114416 x^{3} + 167146 x^{2} + 29932 x + 2471 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-210126339255361190328405271099=-\,7^{12}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.56$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{10} + \frac{3}{7} a^{9} + \frac{1}{7} a^{8} + \frac{3}{7} a^{7} + \frac{3}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{3}$, $\frac{1}{7} a^{12} + \frac{2}{7} a^{10} - \frac{2}{7} a^{9} + \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3}$, $\frac{1}{7} a^{13} + \frac{3}{7} a^{10} + \frac{3}{7} a^{9} + \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{3}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{14} + \frac{1}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{5} + \frac{3}{7} a^{4} - \frac{2}{7} a^{3}$, $\frac{1}{21} a^{15} + \frac{1}{21} a^{14} + \frac{1}{21} a^{13} + \frac{1}{21} a^{12} - \frac{1}{21} a^{11} - \frac{10}{21} a^{10} - \frac{1}{21} a^{9} - \frac{4}{21} a^{8} - \frac{10}{21} a^{7} + \frac{5}{21} a^{6} - \frac{5}{21} a^{5} + \frac{2}{7} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{794201541} a^{16} - \frac{13284010}{794201541} a^{15} - \frac{36358132}{794201541} a^{14} - \frac{14347465}{794201541} a^{13} + \frac{7362716}{264733847} a^{12} + \frac{39142732}{794201541} a^{11} - \frac{115255169}{794201541} a^{10} + \frac{327835051}{794201541} a^{9} + \frac{172416814}{794201541} a^{8} - \frac{17984276}{113457363} a^{7} - \frac{705621}{37819121} a^{6} + \frac{7061740}{113457363} a^{5} + \frac{24169363}{113457363} a^{4} + \frac{105854072}{794201541} a^{3} - \frac{7552388}{37819121} a^{2} - \frac{16524719}{113457363} a - \frac{18394798}{113457363}$, $\frac{1}{19092711683641792184786862999249} a^{17} - \frac{5266869186021441414253}{19092711683641792184786862999249} a^{16} + \frac{24061280403732953091341687893}{19092711683641792184786862999249} a^{15} - \frac{495505027140356141615863218620}{19092711683641792184786862999249} a^{14} + \frac{593952000376539081059433900086}{19092711683641792184786862999249} a^{13} - \frac{442030237058995702577148896891}{6364237227880597394928954333083} a^{12} - \frac{783809025212315417605822540726}{19092711683641792184786862999249} a^{11} - \frac{7292378789324272225380021250816}{19092711683641792184786862999249} a^{10} + \frac{5034129616760167204573530820607}{19092711683641792184786862999249} a^{9} - \frac{9290854386133929389716622945389}{19092711683641792184786862999249} a^{8} + \frac{1349159865658371705354516387550}{19092711683641792184786862999249} a^{7} + \frac{4255028071504164720133066433822}{19092711683641792184786862999249} a^{6} + \frac{2545416167888240472208345769266}{6364237227880597394928954333083} a^{5} - \frac{1619146187137263974829633915046}{19092711683641792184786862999249} a^{4} - \frac{2931191166283424985791773260433}{19092711683641792184786862999249} a^{3} + \frac{29578567691760421981320556536}{909176746840085342132707761869} a^{2} - \frac{104749766483628822822641259405}{909176746840085342132707761869} a + \frac{1222789196397993321934554171442}{2727530240520256026398123285607}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 898497.710099 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-19}) \), 3.1.931.1 x3, 3.3.361.1, 6.0.16468459.2, 6.0.2476099.1, 6.0.5945113699.3 x2, 9.3.105163116221611.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 sibling: 6.0.5945113699.3
Degree 9 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
$19$19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$